From Eq. 13, we see that \(\epsilon \delta H\)  is of the order of \(\epsilon \Delta k^2n_1^2\) .
The magnitude of \(\gamma _{l,m}\)  in Eq. 22 is determined by \(l\)  such factors in the numerator and \(l-1\)  factors involving propagation constant differences: \(\beta ^2_{l^{\prime },m^{\prime }}-\beta ^2_{l^{\prime }\pm 2,m^{\prime \prime }}\approx 2kn_1(\beta _{l^{\prime },m^{\prime }}-\beta _{l^{\prime }\pm 2,m^{\prime \prime }})\) . We now recall that the total number of \((l,m)\)  pairs in a fiber (not counting polarization states and the degenerate counterparts) is approximately \(\alpha V^2/8\)  (for large \(V\) ) [1]}, [2]}, where \(\alpha \)  varies from 0.5 for a parabolic fiber to 1 for a step-index fiber. This implies an average mode spacing of \(k(n_1-n_2)/(\alpha V^2/8)\)  that can for practical purposes for few mode (small \(V\) ) as well as multimode (large \(V\) ) fibers be represented approximately by \(k(n_1-n_2)/V\) ; \(V\)  is the normalized frequency equal to \(ka(n_1^2-n_2^2)^{1/2}=kan_1(2\Delta )^{1/2}\) . Thus,
\(\gamma _{l,m} \sim (\epsilon \Delta k^2 n_1^2)^l/(2k\color {black}^2\color {black}n_1(n_1-n_2)/V)^{l-1}= \Delta k^2 n_1^2 \epsilon ^l (V/2)^{l-1}\) .
Imposing the condition \(\delta \beta ^{2(SO)}_{(l,m)}<< \gamma _{l,m}\)  now leads to the inequality
\(\Big (\frac{l}{k^2a^2n_1^2(V/2)^{l-1}}\Big ) << \epsilon ^l.\) 
